Periodic solutions of nonlinear fractional pantograph integro-differential equations with Ψ-documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Psi -$$end{document}Caputo derivative

摘要

Abstract The aim purpose of the present work is to study the existence and uniqueness of periodic solutions for a wide class of nonlinear fractional integro-differential equations of pantograph type involving Ψ-documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Psi -$$end{document}Caputo derivative operator. The coincidence degree theory introduced before by Mawhin is used for the existence and uniqueness of our problem. The validity of the findings is verified by an appropriate example.